Cofiniteness of local cohomology modules and subcategories of modules

TL;DR

This paper investigates conditions under which local cohomology modules are $I$-cofinite, establishing that if certain modules form an abelian category, then all local cohomology modules of finitely generated modules are also $I$-cofinite.

Contribution

It proves that the $I$-cofiniteness of local cohomology modules extends from the ring to all finitely generated modules under specific categorical assumptions.

Findings

If all $H_I^i(R)$ are $I$-cofinite, then $H_I^i(M)$ are $I$-cofinite for all finitely generated modules $M$.

The $I$-cofinite modules form an abelian category implies the $I$-cofiniteness of local cohomology modules.

The result applies when $R_rak{p}$ is regular local for primes not containing $I$.

Abstract

Let $R$ be a commutative noetherian ring and $I$ an ideal of $R$. Assume that for all integers $i$ the local cohomology module $H_I^i(R)$ is $I$-cofinite. Suppose that $R_\mathfrak{p}$ is a regular local ring for all prime ideals $\mathfrak{p}$ that do not contain $I$. In this paper, we prove that if the $I$-cofinite modules forms an abelian category, then for all finitely generated $R$-modules $M$ and all integers $i$, the local cohomology module $H_I^i(M)$ is $I$-cofinite.